Table Of ContentAtlantis Studies in Differential Equations
Series Editor: Michel Chipot
Stanislav Antontsev
Sergey Shmarev
Evolution PDEs
with Nonstandard
Growth Conditions
Existence, Uniqueness, Localization,
Blow-up
Atlantis Studies in Differential Equations
Volume 4
Series editor
Michel Chipot, Zürich, Switzerland
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Stanislav Antontsev Sergey Shmarev
(cid:129)
Evolution PDEs with
Nonstandard Growth
Conditions
Existence, Uniqueness, Localization,
Blow-up
Stanislav Antontsev Sergey Shmarev
Center forMathematics and Fundamental Department of Mathematics
Applications Universityof Oviedo
Universityof Lisbon Oviedo,Asturias
Lisbon Spain
Portugal
ISSN 2214-6253 ISSN 2214-6261 (electronic)
Atlantis Studies inDifferential Equations
ISBN 978-94-6239-111-6 ISBN 978-94-6239-112-3 (eBook)
DOI 10.2991/978-94-6239-112-3
LibraryofCongressControlNumber:2015935210
©AtlantisPressandtheauthor(s)2015
Thisbook,oranypartsthereof,maynotbereproducedforcommercialpurposesinanyformorbyany
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Printedonacid-freepaper
This work is dedicated to our families,
Tamara, Nikolay, Stanislav and
Elena, Dmitry, Andrey,
to whom we owe so much.
Preface
Thismonographisacontributiontothetheoryofsecondorderquasilinearparabolic
and hyperbolic equations with the nonlinear structure that may change from one
pointtoanotherintheproblemdomain.Inthepastdecade,therewasanimpetuous
growth ofinterest inthestudy ofsuchequations,whichappear inanatural wayin
the mathematical modeling of various real-world phenomena and give rise to
challenging mathematical problems. The aim of this work is to give an account
of the known results on existence, uniqueness, and qualitative properties of
solutions.
The parabolic equations studied below can be conventionally divided into sev-
eral groups. Chaps. 2 and 3 are devoted to study the generalized porous medium
equation
(cid:1) (cid:3)
ut ¼div jujmðx;tÞru þfðx;tÞ ð1Þ
withagivenexponentmðx;tÞ[ (cid:2)1anditsgeneralizations,suchasequationswith
lower order terms or anisotropic equations. We establish conditions of existence
anduniquenessofweaksolutionsandshowthatfordefiniterangesoftheexponent
mðx;tÞ the solutions exhibit properties typical for the solutions of equations with
constant m, those of the finite speed of propagation and extinction in finite time.
The former means the following: if the support of the initial data is compact, then
the support of the solution remains compact for all time but may expand in space
with finite speed. The latter property means that the solution corresponding to a
nonzero initial datum may extinct in a finite time.
Chapters 4–6 concern the homogeneous Dirichlet problem for the nonlinear
degenerate parabolic equations
ut(cid:2)divAðx;t;u;ruÞþBðx;t;uÞ¼0 ð2Þ
vii
viii Preface
with the function A(cid:3)ðA1;...;AnÞ whose components are of the form
Aðx;t;r;ξÞ¼aðx;t;rÞjξjpiðx;tÞ(cid:2)2ξ 8r 2R;ξ2Rn; ð3Þ
i i i i
where piðx;tÞ2ð1;1Þ are given measurable functions and the coefficients ai are
Carathéodoryfunctions,ai 2½a0;a1(cid:4)withpositiveconstantsa0;a1.ThefunctionB
is assumed to satisfy the growth condition jBðx;t;rÞj(cid:5)djrjλþfðx;tÞ with con-
stants d(cid:6)0, λ[1 and a given function f. Special attention is paid to the model
case when
Bðx;t;rÞr ¼cðx;tÞjrjσðx;tÞ(cid:2)fðzÞr 8r 2R ð4Þ
with a continuous exponent σðx;tÞ2ð1;1Þ and a given coefficient cðx;tÞ. Most
of the results remain true if the operators Ai are substituted by the Leray-Lions
operators with variable coercivity and growth conditions.
TheassumptionsonthefunctionsAishowthatinthecaseofvariableexponents
pi thereisagapbetweenthecoercivityandgrowthconditions;forthisreasonsuch
equationsareoftentermedPDEswithnonstandardgrowth.Weextendthisnameto
PDEs of the type (1), to PDEs with anisotropic but possibly constant nonlinearity
and to equations with lower order terms of variable growth. All these equations
share the same important property: they are not scaling-invariant, which makes
inapplicable many of the traditional methods and requires new approaches to the
study of their solvability and the analysisof the qualitative properties of solutions.
The main results of Chap. 4 are the theorems of existence and uniqueness of
weak solutions. The natural analytic framework for the study is furnished by the
Lebesgue and Sobolev spaces with variable exponents which are introduced in
Chap. 1. This chapter collects all the information about the properties of these
spaces used throughout the text. The constructed weak solutions are the so-called
energy solutions which can be taken for the test-function in the corresponding
integral identity. Chapter 4 contains results on the dependence of the regularity
of the weak energy solutions on the regularity of the data and the nonlinearity
exponents pi and σ, and on the unique solvability of the Cauchy problem for the
evolution pðxÞ-Laplace equation. In the final section we provide a review of other
results and approaches to the study of parabolic equations of the type (2).
ThepropertiesofspacelocalizationarestudiedinChap.5.Thestudyisconfined
to the energy solutions of Eq. (2) with the lower order terms of the form (4) with
cðx;tÞ(cid:6)0.Weshowthattherearetwodifferentmechanismswhichcangiveriseto
thepropertyoffinitespeedofpropagation.Thefirstoneisduetoasuitablebalance
between the diffusion and absorption part of the equation. This is possible only if
cðx;tÞ(cid:6)c [0 and is expressed in terms of conditions on the variable rates of
0
growth of the functions Ai and B. The other is caused by the anisotropy of the
diffusion operator and works even in the case when cðx;tÞ(cid:3)0. It turns that if the
anisotropic Eq. (2) combines the directions of slow and fast diffusion, pi[2 or
Preface ix
pi 2ð1;2Þrespectively,thedisturbancesfromthedatarunonlyafiniteorevenzero
distance in the direction of the slowest diffusion.
Chapter6isdevotedtothestudyofthelargetimebehaviorandthephenomenon
of vanishing in a finite time for energy weak solutions of the homogeneous
DirichletproblemforEq.(2).ItisassumedthatthelowerordertermBsatisfies(4)
with cðx;tÞ(cid:6)0. The study is based on the analysis of behavior of the local energy
functions,whichsatisfynonlinearordinarydifferentialequations.Theeffectoftotal
vanishinginfinitetimeisprovidedbyasuitablerelationbetweentheexponentspi,
σ in conditions (3), (4). Besides, the same effect is possible in several situations
specific for equations with the nonstandard growth conditions. It turns out that a
solutionmayvanishinafinitetimeeveniftheequationeventuallytransformsinto
the linear heat equation, which does not admit localized solutions. Extinction in a
finitetimeispossiblealsointhecasewhenthecoefficientcisallowedtovanishon
a set of zero measure. Another effect is due to anisotropy: the equations of
anisotropic diffusion admit solutions localized simultaneously in space and time.
In Chap. 7 we derive conditions of nonexistence of global in time bounded
solutions to Eq. (2). We prove that under certain conditions on the data every
energy solution becomes infinite in a finite time: there exists a finite t(cid:7) such that
kuð(cid:8);tÞk !1 ast !t(cid:7). The following two versions ofEq. (2)are considered:
1;Ω
thesemilinearequationwithAðx;t;ruÞ¼ruandBsatisfyingcondition(4)with
superlinear growth and cðx;tÞ\0, and the quasilinear equation with the exponents
pi andσindependentoft.Theresultsareextendedtononlocalequations,equations
thatbecomelinearastgrowstoinfinity,equationswithnonnegativebutnotstrictly
positive coefficient cðx;tÞ.
Parabolic equations with double degeneracy are studied in Chaps. 8–10.
Chapter 8isdevotedtostudythehomogeneousDirichletproblemfortheisotropic
equations
(cid:1) (cid:3)
ut ¼div ajujαðx;tÞjrujpðx;tÞ(cid:2)2ru þfðx;tÞ:
It is shown that under suitable conditions on the regularity of p and α this
problem has a weak energy solution. Under additional restrictions on the data, the
uniqueness and comparison theorems are proven for the solutions that possess
better regularityandsatisfy theinclusionotu2L1ðQÞ.Itisshownthatthisclassis
nonempty and every weak solution falls into it, provided that the initial data meet
additional regularity assumptions.
Anisotropic doubly degenerate equations are studied in Chaps. 9, 10. We con-
sider the homogeneous Dirichlet problem for the equation
otΨðx;t;uÞ¼divAðx;t;ruÞþcjujσðx;tÞ(cid:2)2uþf ð5Þ
with Ψ ¼jujmðx;tÞ(cid:2)2u, given exponents m, σ and the anisotropic operator A that
satisfiesconditions(3).Iftheexponentspi andmwereconstantandtheoperatorA
wereisotropic,thisequationwouldbeformallyequivalenttotheequationstudiedin
x Preface
Chap.7,butinthecaseofEq.(5)suchareductionisimpossibleandanindependent
analysisisrequired.Chapter9isdevotedtoprovetheexistenceofstrongsolutions
ofEq.(5)whichpossesstheextraregularityproperty:mjujm(cid:2)2u2 2L1ðQÞ.Sufficient
t
conditions for local and global in time existence of bounded strong solutions are
provenandtheenergyrelationsarederived.UnlikethecaseofweaksolutionstoEq.
(2)theproofofexistenceofstrongsolutionstoEq.(5)requirescertainmonotonicity
properties ofthe exponentspi,σ, mand thecoefficientsai and cðx;tÞ.
The energy relations derived in Chap. 9 are used in Chap. 10 to study the
phenomenon of extinction in a finite time, the large time behavior and the possi-
bility of a finite time blow-up for strong solutions of Eq. (5).
In Chaps. 11, 12 we present results of the study of quasilinear and semilinear
hyperbolic equations with nonstandard growth conditions. The homogeneous
Dirichlet problem for the equation
ut ¼divAðx;t;ruÞþεΔutþcjujσðx;tÞ(cid:2)2uþf
withtheisotropicoperatorAoftheform(3)isstudiedinChap.11.Twodifferent
cases are considered: the equation with the damping term, ε[0, and the equation
with ε¼0. It is shown that for every ε[0 the problem admits global or local in
time weak solutions, provided that the exponents of nonlinearity satisfy certain
regularity assumptions. The solutions of the damped problem may blow-up in a
finitetimeandtheblow-upmomentt(cid:7) admitsthetwo-sidedbounds,whichdepend
ontheproblemdatabutareindependentofε.Ifε¼0,thequestionofexistenceof
weak solutions is left open. Nonetheless, it is proven that in this case the problem
admitstheweakersolutioninthesenseofYoungmeasure,whichisobtainedasthe
limitofthesequenceofweaksolutionstothedampedproblemsasε!0.Stronger
results are obtained in Chap. 12 for the semilinear hyperbolic equation
u ¼Δuþcðx;tÞjujσðx;tÞ(cid:2)2uþf:
tt
We derive sufficient conditions for local in time existence of weak and strong
solutions and prove uniqueness of a strong solution.It isshown that in thecase of
superlineargrowth,thatis,ifσðx;tÞ[2,everynonnegativestrongsolutionblows-
up in a finite time. The nonexistence result is extended to equations with nonlocal
lower order terms, to equations which transform into the linear wave equation as
t!1andtothecasewhenthecoefficientcðx;tÞisnotseparatedawayfromzero
in the problem domain.
The bulk of the presented material is constituted by the original results of the
authorsobtainedincourseofthepast10years.Theotherpertinentresultsscattered
in the literature are reviewed in each chapter. Although the selection and presen-
tation of the supplementary material always reflect the interests of the authors, we
expect that the reader will find them quite complete.
Theauthorsacknowledgethesupportofseveralinstitutionsreceivedatthefinal
stage of the work: the Research Grants CAPES-PVE-88887.059583/2014-00 and
